Properties

Label 2-93058-1.1-c1-0-1
Degree $2$
Conductor $93058$
Sign $1$
Analytic cond. $743.071$
Root an. cond. $27.2593$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 2·12-s + 2·13-s + 14-s − 4·15-s + 16-s − 18-s + 6·19-s + 2·20-s + 2·21-s − 4·22-s + 23-s + 2·24-s − 25-s − 2·26-s + 4·27-s − 28-s + 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.577·12-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.852·22-s + 0.208·23-s + 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.730·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93058\)    =    \(2 \cdot 7 \cdot 17^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(743.071\)
Root analytic conductor: \(27.2593\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{93058} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 93058,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.165505481\)
\(L(\frac12)\) \(\approx\) \(1.165505481\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + T \)
17 \( 1 \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.98146668797855, −13.17107551554964, −12.85866986827634, −12.10661955896789, −11.68859381958844, −11.48995478544696, −10.79347936770802, −10.42215894752268, −9.789240661003470, −9.453274055937247, −9.007735006391055, −8.476883858808427, −7.706258440025943, −7.119970560387393, −6.604818271087886, −6.192093780271551, −5.801177666515502, −5.263995873297917, −4.709311398001852, −3.743622160825724, −3.317474205816516, −2.506041546546641, −1.575899068635131, −1.277931053768455, −0.4461746401343343, 0.4461746401343343, 1.277931053768455, 1.575899068635131, 2.506041546546641, 3.317474205816516, 3.743622160825724, 4.709311398001852, 5.263995873297917, 5.801177666515502, 6.192093780271551, 6.604818271087886, 7.119970560387393, 7.706258440025943, 8.476883858808427, 9.007735006391055, 9.453274055937247, 9.789240661003470, 10.42215894752268, 10.79347936770802, 11.48995478544696, 11.68859381958844, 12.10661955896789, 12.85866986827634, 13.17107551554964, 13.98146668797855

Graph of the $Z$-function along the critical line